Open Access

Fixed Point Iterations of a Pair of Hemirelatively Nonexpansive Mappings

Fixed Point Theory and Applications20102010:270150

DOI: 10.1155/2010/270150

Received: 27 September 2009

Accepted: 22 March 2010

Published: 31 March 2010

Abstract

We introduce an iterative method for a pair of hemirelatively nonexpansive mappings. Strong convergence of the purposed iterative method is obtained in a Banach space.

1. Introduction and Preliminaries

Let be a Banach space with the dual . We denote by the normalized duality mapping from to defined by

(1.1)

where denotes the generalized duality pairing. A Banach space is said to be strictly convex if for all with and It is said to be uniformly convex if for any two sequences in such that and . Let be the unit sphere of . Then the Banach space is said to be smooth provided that

(1.2)

exists for each It is also said to be uniformly smooth if the limit (1.2) is attained uniformly for . It is well known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of . It is also well known that is uniformly smooth if and only if is uniformly convex.

Recall that a Banach space has the Kadec-Klee property if for any sequences and with and then as ; for more details on Kadec-Klee property, the readers is referred to [1, 2] and the references therein. It is well known that if is a uniformly convex Banach space, then enjoys the Kadec-Klee property.

Let be a nonempty closed and convex subset of a Banach space and  :   →  a mapping. The mapping is said to be closed if for any sequence such that and , then . A point is a fixed point of provided . In this paper, we use to denote the fixed point set of and use and to denote the strong convergence and weak convergence, respectively.

Recall that the mapping is said to be nonexpansive if

(1.3)

It is well known that if is a nonempty bounded closed and convex subset of a uniformly convex Banach space , then every nonexpansive self-mapping on has a fixed point. Further, the fixed point set of is closed and convex.

As we all know that if is a nonempty closed convex subset of a Hilbert space and  :   →  is the metric projection of onto , then is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [3] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

Next, we assume that is a smooth Banach space. Consider the functional defined by

(1.4)

Observe that, in a Hilbert space , (1.4) is reduced to The generalized projection  :   →  is a map that assigns to an arbitrary point , the minimum point of the functional that is, where is the solution to the minimization problem

(1.5)

Existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping (see, e.g., [14]). In Hilbert spaces, It is obvious from the definition of function that

(1.6)

Remark 1.1.

If is a reflexive, strictly convex and smooth Banach space, then for , if and only if . It is sufficient to show that if then . From (1.6), we have . This implies that From the definition of we have . Therefore, we have see [1, 2] for more details.

Let be a nonempty closed convex subset of and a mapping from into itself. A point in is said to be an asymptotic fixed point of [5] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is said to be relatively nonexpansive [3, 6, 7] if and for all and . The mapping is said to be hemirelatively nonexpansive [812] if and for all and . The asymptotic behavior of a relatively nonexpansive mappings was studied in [3, 6, 7].

Remark 1.2.

The class of hemirelatively nonexpansive mappings is more general than the class of relatively nonexpansive mappings which requires the restriction: . From Su et al. [11], we see that every hemirelatively nonexpansive mapping is relatively nonexpansive, but the inverse is not true. Hemirelatively nonexpansive mapping is also said to be quasi- -nonexpansive; see [1317].

Recently, fixed point iterations of relatively nonexpansive mappings and hemirelatively nonexpansive mappings have been considered by many authors; see, for example [1425] and the references therein. In 2005, Matsushita and Takahashi [8] considered fixed point problems of a single relatively nonexpansive mapping in a Banach space. To be more precise, they proved the following theorem.

Theorem 1 MT.

Let be a uniformly convex and uniformly smooth Banach space; let be a nonempty closed convex subset of let be a relatively nonexpansive mapping from into itself; let be a sequence of real numbers such that and . Suppose that is given by
(1.7)

where is the duality mapping on . If is nonempty, then converges strongly to , where is the generalized projection from onto

In 2007, Plubtieng and Ungchittrakool [9] further improved Theorem MT by considering a pair of relatively nonexpansive mappings. To be more precise, they proved the following theorem.

Theorem PU

Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of Let and be two relatively nonexpansive mappings from into itself with being nonempty. Let a sequence be defined by
(1.8)

with the following restrictions:

for each and

for each , and .

Then the sequence converges strongly to , where is the generalized projection from onto

Very recently, Su et al. [11] improved Theorem PU partially by considering a pair of hemirelatively nonexpansive mappings. To be more precise, they obtained the following results.

Theorem SWX

Let be a uniformly convex and uniformly smooth Banach space, and let be a nonempty closed convex subset of Let and be two closed hemirelatively nonexpansive mappings from into itself with being nonempty. Let a sequence be defined by
(1.9)

with the following restrictions:

(1) ;

(2) ;

(3) for some .

Then the sequence converges strongly to , where is the generalized projection from onto

In this paper, motivated by Theorems MT, PU, and SWX, we consider the problem of finding a common fixed point of a pair of hemirelatively nonexpansive mappings by shrinking projection methods which were introduced by Takahashi et al. [26] in Hilbert spaces. Strong convergence theorems of common fixed points are established in a Banach space. The results presented in this paper mainly improve the corresponding results announced in Matsushita and Takahashi [8], Nakajo and Takahahsi [27], and Su et al. [11].

In order to prove our main results, we need the following lemmas.

Lemma 1.3 (see [3]).

Let be a nonempty closed convex subset of a smooth Banach space and . Then, if and only if
(1.10)

Lemma 1.4 (see [3]).

Let be a reflexive, strictly convex and smooth Banach space, a nonempty closed convex subset of , and Then
(1.11)

The following lemma can be deduced from Matsushita and Takahashi [8].

Lemma 1.5.

Let be a strictly convex and smooth Banach space, a nonempty closed convex subset of and a hemirelatively nonexpansive mapping. Then is a closed convex subset of .

Lemma 1.6 (see [28]).

Let be a uniformly convex Banach space and a closed ball of Then there exists a continuous strictly increasing convex function with such that
(1.12)

for all and with

2. Main Results

Theorem 2.1.

Let be a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property and a nonempty closed and convex subset of Let and be two closed and hemirelatively nonexpansive mappings such that is nonempty. Let be a sequence generated in the following manner:
(2.1)

where , , , and are real sequences in satisfying the following restrictions:

and .

Then converges strongly to , where is the generalized projection from onto .

Proof.

First, we show that is closed and convex for each It is obvious that is closed and convex. Suppose that is closed and convex for some . For , we see that is equivalent to
(2.2)
It is easy to see that is closed and convex. Then, for each , is closed and convex. Now, we are in a position to show that for each Indeed, is obvious. Suppose that for some . Then, for all , we have
(2.3)
It follows that
(2.4)
which shows that . This implies that for each On the other hand, we obtain from Lemma 1.4 that
(2.5)
for each and for each This shows that the sequence is bounded. From (1.6), we see that the sequence is also bounded. Since the space is reflexive, we may, without loss of generality, assume that . Note that is closed and convex for each . It is easy to see that for each Note that
(2.6)
It follows that
(2.7)
This implies that
(2.8)

Hence, we have as In view of the Kadec-Klee property of we obtain that as

Next, we show that By the construction of we have that and It follows that
(2.9)
Letting in (2.9), we obtain that . In view of , we arrive at It follows that
(2.10)
From (1.6), we can obtain that
(2.11)
It follows that
(2.12)
This implies that is bounded. Note that is reflexive and is also reflexive. We may assume that In view of the reflexivity of , we see that This shows that there exists an such that It follows that
(2.13)
Taking , the both sides of equality above yield that
(2.14)
That is, which in turn implies that It follows that From (2.12) and since enjoys the Kadec-Klee property, we obtain that
(2.15)
Note that is demicontinuous. It follows that From (2.11) and since enjoys the Kadec-Klee property, we obtain that
(2.16)
Note that
(2.17)
It follows that
(2.18)
Since is uniformly norm-to-norm continuous on any bounded sets, we have
(2.19)
On the other hand, we see from the definition of that
(2.20)
In view of the assumption on and (2.19), we see that
(2.21)
On the other hand, since is demicontinuous, we have In view of
(2.22)
we arrive at as By virtue of the Kadec-Klee property of , we obtain that as Note that
(2.23)
In view of (2.21), we arrive at Since is demicontinuous, we have Note that
(2.24)
It follows that as . Since enjoys the Kadec-Klee property, we obtain that Note that
(2.25)
It follows that
(2.26)
Let . Fixing we have from Lemma 1.6 that
(2.27)
It follows that
(2.28)
On the other hand, we have
(2.29)
It follows from (2.21) and (2.26) that
(2.30)
In view of (2.28) and the assumption , we see that
(2.31)
It follows from the property of that
(2.32)
Note that
(2.33)
On the other hand, we have
(2.34)
From (2.32) and (2.33), we arrive at
(2.35)
Note that is demicontinuous. It follows that On the other hand, we have
(2.36)
In view of (2.35), we obtain that as . Since enjoys the Kadec-Klee property, we obtain that
(2.37)

It follows from the closedness of that By repeating (2.27)–(2.37), we can obtain that . This shows that

Finally, we show that From we have
(2.38)
Taking the limit as in (2.38), we obtain that
(2.39)

and hence by Lemma 1.3. This completes the proof.

Remark 2.2.

Theorem 2.1 improves Theorem SWX in the following aspects:

from the point of view on computation, we remove the set in Theorem SWX;

from the point of view on the framework of spaces, we extend Theorem SWX from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property. Note that every uniformly convex Banach space enjoys the Kadec-Klee property.

If for each , then Theorem 2.1 is reduced to the following.

Corollary 2.3.

Let be a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property and a nonempty closed and convex subset of Let  :  and  :  be two closed and hemirelatively nonexpansive mappings such that is nonempty. Let be a sequence generated in the following manner:
(2.40)

where , , and are real sequences in satisfying the following restrictions:

and .

Then converges strongly to , where is the generalized projection from onto .

If , then Corollary 2.3 is reduced to the following.

Corollary 2.4.

Let be a uniformly smooth and strictly convex Banach space which enjoys the Kadec-Klee property and a nonempty closed and convex subset of . Let be a closed and hemirelatively nonexpansive mapping with a nonempty fixed point set. Let be a sequence generated in the following manner:
(2.41)

where is a real sequence in satisfying . Then converges strongly to , where is the generalized projection from onto .

Declarations

Acknowledgment

This project is supported by the National Natural Science Foundation of China (no. 10901140).

Authors’ Affiliations

(1)
School of Mathematics Physics and Information Science, Zhejiang Ocean University
(2)
College of Mathematics, Physics and Information Engineering, Zhejiang Normal University
(3)
Department of Mathematics, Gyeongsang National University

References

  1. Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Mathematics and Its Applications. Volume 62. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1990:xiv+260.View ArticleMATHGoogle Scholar
  2. Takahashi W: Nonlinear Functional Analysis, Fixed Point Theory and Its Application. Yokohama, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
  3. Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Edited by: Kartsatos AG. Marcel Dekker, New York, NY, USA; 1996:15–50.Google Scholar
  4. Alber YaI, Reich S: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panamerican Mathematical Journal 1994,4(2):39–54.MathSciNetMATHGoogle Scholar
  5. Reich S: A weak convergence theorem for the alternating method with Bregman distances. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Edited by: Kartsatos AG. Marcel Dekker, New York, NY, USA; 1996:313–318.Google Scholar
  6. Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. Journal of Applied Analysis 2001,7(2):151–174. 10.1515/JAA.2001.151MathSciNetView ArticleMATHGoogle Scholar
  7. Butnariu D, Reich S, Zaslavski AJ: Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numerical Functional Analysis and Optimization 2003,24(5–6):489–508. 10.1081/NFA-120023869MathSciNetView ArticleMATHGoogle Scholar
  8. Matsushita SY, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. Journal of Approximation Theory 2005,134(2):257–266. 10.1016/j.jat.2005.02.007MathSciNetView ArticleMATHGoogle Scholar
  9. Plubtieng S, Ungchittrakool K: Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space. Journal of Approximation Theory 2007,149(2):103–115. 10.1016/j.jat.2007.04.014MathSciNetView ArticleMATHGoogle Scholar
  10. Qin X, Su Y: Strong convergence theorems for relatively nonexpansive mappings in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2007,67(6):1958–1965. 10.1016/j.na.2006.08.021MathSciNetView ArticleMATHGoogle Scholar
  11. Su Y, Wang Z, Xu H: Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(11):5616–5628. 10.1016/j.na.2009.04.053MathSciNetView ArticleMATHGoogle Scholar
  12. Su Y, Wang D, Shang M: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, 2008:-8.Google Scholar
  13. Qin X, Cho YJ, Kang SM, Zhou H: Convergence of a modified Halpern-type iteration algorithm for quasi--nonexpansive mappings. Applied Mathematics Letters 2009,22(7):1051–1055. 10.1016/j.aml.2009.01.015MathSciNetView ArticleMATHGoogle Scholar
  14. Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. Journal of Computational and Applied Mathematics 2009,225(1):20–30. 10.1016/j.cam.2008.06.011MathSciNetView ArticleMATHGoogle Scholar
  15. Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi--nonexpansive mappings and equilibrium problems. Journal of Computational and Applied Mathematics 2010, 234: 750–760. 10.1016/j.cam.2010.01.015MathSciNetView ArticleMATHGoogle Scholar
  16. Qin X, Cho YJ, Cho SY, Kang SM: Strong convergence theorems of common fixed points for a family of quasi--nonexpansive mappings. Fixed Point Theory and Applications 2010, 2010:-11.Google Scholar
  17. Zhou H, Gao G, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi--asymptotically nonexpansive mappings. Journal of Applied Mathematics and Computing 2010, 32: 453–464. 10.1007/s12190-009-0263-4MathSciNetView ArticleMATHGoogle Scholar
  18. Kimura Y, Takahashi W: On a hybrid method for a family of relatively nonexpansive mappings in a Banach space. Journal of Mathematical Analysis and Applications 2009,357(2):356–363. 10.1016/j.jmaa.2009.03.052MathSciNetView ArticleMATHGoogle Scholar
  19. Lewicki G, Marino G: On some algorithms in Banach spaces finding fixed points of nonlinear mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,71(9):3964–3972. 10.1016/j.na.2009.02.066MathSciNetView ArticleMATHGoogle Scholar
  20. Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi--nonexpansive mappings. Applied Mathematics and Computation 2010,215(11):3874–3883. 10.1016/j.amc.2009.11.031MathSciNetView ArticleMATHGoogle Scholar
  21. Qin X, Su Y: Approximation of a zero point of accretive operator in Banach spaces. Journal of Mathematical Analysis and Applications 2007,329(1):415–424. 10.1016/j.jmaa.2006.06.067MathSciNetView ArticleMATHGoogle Scholar
  22. Qin XL, Cho YJ, Kang SM, Zhou HY: Convergence of a hybrid projection algorithm in Banach spaces. Acta Applicandae Mathematicae 2009,108(2):299–313. 10.1007/s10440-008-9313-4MathSciNetView ArticleMATHGoogle Scholar
  23. Wei L, Cho YJ, Zhou H: A strong convergence theorem for common fixed points of two relatively nonexpansive mappings and its applications. Journal of Applied Mathematics and Computing 2009,29(1–2):95–103. 10.1007/s12190-008-0092-xMathSciNetView ArticleMATHGoogle Scholar
  24. Wattanawitoon K, Kumam P: Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings. Nonlinear Analysis: Hybrid Systems 2009,3(1):11–20. 10.1016/j.nahs.2008.10.002MathSciNetMATHGoogle Scholar
  25. Zegeye H, Shahzad N: Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 2009,70(7):2707–2716. 10.1016/j.na.2008.03.058MathSciNetView ArticleMATHGoogle Scholar
  26. Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062MathSciNetView ArticleMATHGoogle Scholar
  27. Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4MathSciNetView ArticleMATHGoogle Scholar
  28. Cho YJ, Zhou H, Guo G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Computers & Mathematics with Applications 2004,47(4–5):707–717. 10.1016/S0898-1221(04)90058-2MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Y. Hao and S. Y. Cho. 2010

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